Hermitian variation of an almost pseudo-Kähler metric

Given an almost pseudo-Kähler manifold and a vector field we construct an almost pseudo-Hermitian metric. We study under which conditions this new pseudo-Hermitian metric is also almost pseudo-Kähler and we use it to get some information about holomorphic vector fields on a compact pseudo-Kähler manifold with index 2. We also study the relationship between holomorphic statistical structures and the newly constructed pseudo-Hermitian metric.